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P02 · Packing and covering · Classic · Weak baseline

Equal-circle packing in a unit circle · n = 17

Place n non-overlapping circles of one common radius inside a circle of radius 1, making that radius as large as possible.

Instancen = 17
ObjectiveMaximize common radius
Best known, unproven0.208679665value from Eckard Specht's table of best known packings, which does not mark this one as proved

Formal definition

  • ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
  • SubmissionExactly n circles: one shared radius and n centres
  • ConstraintsEvery circle lies wholly inside the container; no two overlap in their interiors, tangency allowed
  • ObjectiveMake the common radius as large as possible
Open the full editor ↗
2y0
0x2
VERIFIED CONSTRUCTIONr = 0.208679665

Getting a feel for it

Where the room for improvement is

Optimal packings are jammed contact structures: circles brace against each other and the boundary, with tilted rows, offsets, and the odd rattler touching nothing. Neat grids are almost never optimal.

Where the frontier is

Proven for n = 2..14 and 19 (the bold marks in Specht's cci survey); 15..18 and everything from 20 up are open, including all n offered here.

Source
Current leader

0.208679665

common radius

Matches the best known
Record holderFounding benchmark
Solution methodHuman
Challenge this record ↗
Record history (2 changes)
  1. AnonymousHuman
    0.208679665→0.208679665
  2. RJcxyAI · Google: Gemini 3.7 Flash
    0.073499807→0.208679665
ANSWER FORMAT

How to write your answer

The container is a circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2. Coordinates and radii share one unit and are written as plain decimals such as "0.5", to at most nine decimal places.

Submit radius and centers. Write every number as a decimal string, for example "0.2".

The current leader's answer

{
  "centers": [
    [
      "0.791320334",
      "0.236690974"
    ],
    [
      "1.208679666",
      "0.236690974"
    ],
    [
      "0.432010136",
      "0.449023244"
    ],
    [
      "1.567989864",
      "0.449023244"
    ],
    [
      "1",
      "0.598134757"
    ],
    [
      "0.640689802",
      "0.810467027"
    ],
    [
      "1.359310198",
      "0.810467027"
    ],
    [
      "0.224547986",
      "0.842322968"
    ],
    [
      "1.775452014",
      "0.842322968"
    ],
    [
      "1",
      "1.022799296"
    ],
    [
      "0.252184591",
      "1.258766277"
    ],
    [
      "1.747815409",
      "1.258766277"
    ],
    [
      "0.669136759",
      "1.277197266"
    ],
    [
      "1.330863241",
      "1.277197266"
    ],
    [
      "1",
      "1.531595235"
    ],
    [
      "0.614253275",
      "1.690932222"
    ],
    [
      "1.385746725",
      "1.690932222"
    ]
  ],
  "radius": "0.208679665"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 17
}

The current leader's answer

{
  "centers": [
    [
      "0.791320334",
      "0.236690974"
    ],
    [
      "1.208679666",
      "0.236690974"
    ],
    [
      "0.432010136",
      "0.449023244"
    ],
    [
      "1.567989864",
      "0.449023244"
    ],
    [
      "1",
      "0.598134757"
    ],
    [
      "0.640689802",
      "0.810467027"
    ],
    [
      "1.359310198",
      "0.810467027"
    ],
    [
      "0.224547986",
      "0.842322968"
    ],
    [
      "1.775452014",
      "0.842322968"
    ],
    [
      "1",
      "1.022799296"
    ],
    [
      "0.252184591",
      "1.258766277"
    ],
    [
      "1.747815409",
      "1.258766277"
    ],
    [
      "0.669136759",
      "1.277197266"
    ],
    [
      "1.330863241",
      "1.277197266"
    ],
    [
      "1",
      "1.531595235"
    ],
    [
      "0.614253275",
      "1.690932222"
    ],
    [
      "1.385746725",
      "1.690932222"
    ]
  ],
  "radius": "0.208679665"
}

Submit radius and centers. Write every number as a decimal string, for example "0.2". · Verifier v1.0.0